For a long time-at least from Fermat to Minkowski-the theory of quadratic forms was a part of
number theory. Much of the best work of the great number theorists of the eighteenth and …

Hilbert's 17th Problem asks whether a real positive semidefinite polynomial can be
expressed as a sum of squares of rational functions. Artin answered “yes” in the 1920's …

In this paper, we define flasque tori and flasque resolutions of tori over an arbitrary base
scheme (Sect. 1) and we establish the basic cohomological properties of flasque tori over a …

Introduction. Real Algebra, roughly speaking, is the study of" real" objects such as real rings,
real places and real varieties. Becuase of the recent interest in developing algebraic …

Let $ V $ be an affine algebraic variety over $\mathbb {R} $(or any other real closed field $ R
$). We ask when it is true that every positive semidefinite (psd) polynomial function on $ V …

A real polynomial pclR [xl,..., x J is called positive semidefinite (psd) if p (al,..., a,)> O for all
real ai (we write p> 0 for short). A form (homogeneous polynomial) in n variables of degree …

For a (commutative) ring A, thepythagoras number, P (A), ofA is the smallest number n^ oo
such that any sum of squares in A can be expressed äs a sum of at most n squares in A. For …

Many classical and modern results and quadratic forms are brought together in this book.
The treatment is self-contained and of a totally elementary nature requiring only a basic …

F was defined to be 1-amenable (resp. strongly 1-amenable) if (1.1) is exact at WF (resp. at
WF and WM) for every multiquadratic extension M of F. It was shown in [ELW1,(3.2)] and …

Let A be a semilocal ring. We compare the set of positive semide® nite (psd) elements of A
and the set of sums of squares in A. For psd fe A, whether f is a sum of squares or not …